Some complexity results in the theory of normal numbers

TL;DR

This paper investigates the descriptive complexity of sets related to normal numbers, showing that certain sets are $oldsymbol{ ext{Pi}}^0_3$-complete and others are at the $oldsymbol{ ext{Pi}}^0_4$ level, with implications for their Hausdorff dimensions.

Contribution

It establishes the exact descriptive complexity of sets of normal numbers and related noise-based sets, demonstrating the limits of possible characterizations.

Findings

The set of normal numbers is $oldsymbol{ ext{Pi}}^0_3$-complete.

The set of numbers preserving normality under addition is also $oldsymbol{ ext{Pi}}^0_3$-complete.

Some related sets are complete at the $oldsymbol{ ext{Pi}}^0_4$ level.

Abstract

Let $\mathscr{N}(b)$ be the set of real numbers which are normal to base $b$. A well-known result of H. Ki and T. Linton is that $\mathscr{N}(b)$ is $\boldsymbol{\Pi}^0_3$-complete. We show that the set $\mathscr{N}(b)$ of reals which preserve $\mathscr{N}(b)$ under addition is also $\boldsymbol{\Pi}^0_3$-complete. We use the characteriztion of $\mathscr{N}(b)$ given by G. Rauzy in terms of an entropy-like quantity called the noise. It follows from our results that no further characteriztion theorems could result in a still better bound on the complexity of $\mathscr{N}(b)$. We compute the exact descriptive complexity of other naturally occurring sets associated with noise. One of these is complete at the $\boldsymbol{\Pi}^0_4$ level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.